Alice, Bob, and Confucius are bored during recess, so they decide to play a new game. Each of them puts a dollar in the pot, and each tosses a coin. Alice wins if the coins land all heads or all tails. Bob wins if two heads and one tail land, and Confucius wins if one head and two tails land. The coins are fair, and the winner receives a net payment of $2 ($3 - $1 = $2), and the losers lose their $1.

Alice, Bob, and Confucius are bored during recess, so they decide to play a new game. Each of them puts a dollar in the pot, and each tosses a coin. Alice wins if the coins land all heads or all tails. Bob wins if two heads and one tail land, and Confucius wins if one head and two tails land. The coins are fair, and the winner receives a net payment of $2 ($3 - $1 = $2), and the losers lose their $1. What is Confucius' expected payoff? Round your answer to three decimal places (e.g. 1.234)

There are n𝑛 coins on the table forming a circle, and each coin is either facing up or facing down. Alice and Bob take turns to play the following game, and Alice goes first.In each operation, the player chooses a facing-up coin, removes the coin, and flips the two coins that are adjacent to it. If there are only two coins left, then one will be removed and the other won't be flipped (as it would be flipped twice). If there is only one coin left, no coins will be flipped. If there are no facing-up coins, the player loses.Decide who will win the game if they both play optimally. It can be proved that the game will end in a finite number of operations, and one of them will win.InputEach test contains multiple test cases. The first line contains the number of test cases t𝑡 (1≤t≤1001≤𝑡≤100). The description of the test cases follows.The first line of each test case contains only one positive integer n𝑛 (1≤n≤1001≤𝑛≤100), representing the number of the coins.A string s𝑠 of length n𝑛 follows on the second line of each test case, containing only "U" and "D", representing that each coin is facing up or facing down.OutputFor each test case, print "YES" if Alice will win the game, and "NO" otherwise.You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.ExampleinputCopy35UUDUD5UDDUD2UUoutputCopyYESNONO

Three balanced coins are flipped independently. One of the variables of interest is X, thenumber of heads. Let Y denote the amount of money won on a side bet in the followingmanner. If the first head occurs on the first flip, you win $1. If the first head occurs onthe second flip you win $2 and if the first head occurs on the third flip you win $3. Ifno heads appear, you lose $1 (that is, you win −$1).(a) [3 marks] In a table, list all possible outcomes of the experiment, along with thevalues of X and Y associated with each outcome.(b) [3 marks] Determine the bivariate distribution (that is, the joint probability dis-tribution) of X and Y . You can list the probabilities in a table.(c) [3 marks] Find the probability that fewer than three heads will occur and you willwin $1 or less.(d) [2 marks] Are X and Y independent? Why or why no

You have been offered to play a game. In this game, there are n𝑛 possible outcomes, and for each of them, you must bet a certain integer amount of coins. In the event that the i𝑖-th outcome turns out to be winning, you will receive back the amount of coins equal to your bet on that outcome, multiplied by ki𝑘𝑖. Note that exactly one of the n𝑛 outcomes will be winning.Your task is to determine how to distribute the coins in such a way that you will come out ahead in the event of any winning outcome. More formally, the total amount of coins you bet on all outcomes must be strictly less than the number of coins received back for each possible winning outcome.InputEach test consists of multiple test cases. The first line contains a single integer t𝑡 (1≤t≤1041≤𝑡≤104) — the number of test cases. The description of the test cases follows.The first line of each test case contains a single integer n𝑛 (1≤n≤501≤𝑛≤50) — the number of outcomes.The second line of each test case contains n𝑛 integers k1,k2,…,kn𝑘1,𝑘2,…,𝑘𝑛 (2≤ki≤202≤𝑘𝑖≤20) — the multiplier for the amount of coins if the i𝑖-th outcome turns out to be winning.It is guaranteed that the sum of n𝑛 over all test cases does not exceed 2⋅1052⋅105.OutputFor each test case, output −1−1 if there is no way to distribute the coins as required. Otherwise, output n𝑛 integers x1,x2,…,xn𝑥1,𝑥2,…,𝑥𝑛 (1≤xi≤1091≤𝑥𝑖≤109) — your bets on the outcomes.It can be shown that if a solution exists, there is always a solution that satisfies these constraints.If there are multiple suitable solutions, output any of them.ExampleinputCopy633 2 723 355 5 5 5 567 9 3 17 9 1336 3 259 4 6 8 3outputCopy27 41 12 1 1 -11989 1547 4641 819 1547 1071 -18 18 12 9 24NoteIn the first test case, the coins can be distributed as follows: 2727 coins on the first outcome, 4141 coins on the second outcome, 1212 coins on the third outcome. Then the total amount of coins bet on all outcomes is 27+41+12=8027+41+12=80 coins. If the first outcome turns out to be winning, you will receive back 3⋅27=813⋅27=81 coins, if the second outcome turns out to be winning, you will receive back 2⋅41=822⋅41=82 coins, if the third outcome turns out to be winning, you will receive back 7⋅12=847⋅12=84 coins. All these values are strictly greater than 8080.In the second test case, one way is to bet one coin on each of the outcomes.

The King wants to marry off his daughter, and he wants her husband to have the greatest innate luckiness possible. To find such a person he decided to hold a heads-or-tails tournament.If person A𝐴 with luckiness x𝑥 and person B𝐵 with luckiness y𝑦 play heads-or-tails against each other, person A𝐴 wins with probability x/(x+y)𝑥/(𝑥+𝑦).The tournament has several rounds. Each round some participants are split into pairs. Each pair plays against each other, and the loser leaves the tournament.The participants are numbered from 11 to n𝑛. During the first round, a number k𝑘 (1≤k≤n1≤𝑘≤𝑛) is selected such that n−k/2𝑛−𝑘/2 is a power of 22 (such k𝑘 always exists and is unique). Only participants numbered from 11 to k𝑘 take part in the first round. It ensures that in all other rounds the number of participants is the power of 22.During other rounds, all the participants who still have not left the tournament take part. If during some round, participants numbered p1<…<p2m𝑝1<…<𝑝2𝑚 take part, then they are split into pairs in the following manner: participant p2i−1𝑝2𝑖−1 plays against participant p2i𝑝2𝑖 for each i𝑖 from 11 to m𝑚.The rounds are held until only one participant is left. He is declared the winner of the tournament and he will marry the King's daughter. The princess can't wait to find out who is her future husband. She asked every participant to tell her his luckiness. Assuming they did not lie, she wants to know the probability of each participant winning the tournament. As you are the best friend of the princess, she asks you to help her.InputThe first line of the input contains the number of participants, n𝑛 (2≤n≤3⋅1052≤𝑛≤3⋅105). The second line of the input contains n𝑛 integer numbers, a1,…,an𝑎1,…,𝑎𝑛 (1≤ai≤1091≤𝑎𝑖≤109). The luckiness of the i𝑖-th participant equals to ai𝑎𝑖.OutputPrint n𝑛 numbers pi𝑝𝑖. The i𝑖-th number should be the probability of the i𝑖-th participant winning the tournament. The absolute error of your answer must not exceed 10−910−9.ExampleinputCopy51 4 1 1 4outputCopy0.026 0.3584 0.0676 0.0616 0.4864NoteHere is an example of a tournament bracket, showing the winning probability in each pair.